Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.     

A permutation test for multivariate data with grouped components

In this paper, we consider a nonparametric test procedure for multivariate data with grouped components under the two sample problem setting. For the construction of the test statistic, we use linear rank statistics which were derived by applying the likelihood ratio principle for each component. For the null distribution of the test statistic, we apply the permutation principle for small or moderate sample sizes and derive the limiting distribution for the large sample case. Also we illustrate our test procedure with an example and compare with other procedures through simulation study. Finally, we discuss some additional interesting features as concluding remarks.     

Tilting methods for assessing the influence of components in a classifier

Summary.  Many contemporary classifiers are constructed to provide good performance for very high dimensional data. However, an issue that is at least as important as good classification is determining which of the many potential variables provide key information for good decisions. Responding to this issue can help us to determine which aspects of the datagenerating mechanism (e.g. which genes in a genomic study) are of greatest importance in terms of distinguishing between populations. We introduce tilting methods for addressing this problem. We apply weights to the components of data vectors, rather than to the data vectors themselves (as is commonly the case in related work). In addition we tilt in a way that is governed by L ₂-distance between weight vectors, rather than by the more commonly used Kullback–Leibler distance. It is shown that this approach, together with the added constraint that the weights should be non-negative, produces an algorithm which eliminates vector components that have little influence on the classification decision. In particular, use of the L ₂-distance in this problem produces properties that are reminiscent of those that arise when L ₁-penalties are employed to eliminate explanatory variables in very high dimensional prediction problems, e.g. those involving the lasso. We introduce techniques that can be implemented very rapidly, and we show how to use bootstrap methods to assess the accuracy of our variable ranking and variable elimination procedures.     

Robust estimation of variance components

New robust estimates for variance components are introduced. Two simple models are considered: the balanced one-way classification model with a random factor and the balanced mixed model with one random factor and one fixed factor. However, the method of estimation proposed can be extended to more complex models. The new method of estimation we propose is based on the relationship between the variance components and the coefficients of the least-mean-squared-error predictor between two observations of the same group. This relationship enables us to transform the problem of estimating the variance components into the problem of estimating the coefficients of a simple linear regression model. The variance-component estimators derived from the least-squares regression estimates are shown to coincide with the maximum-likelihood estimates. Robust estimates of the variance components can be obtained by replacing the least-squares estimates by robust regression estimates. In particular, a Monte Carlo study shows that for outlier-contaminated normal samples, the estimates of variance components derived from GM regression estimates and the derived test outperform other robust procedures.     

Further results for parallel systems with two heterogeneous exponential components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here as well.     

Bayesian multivariate Poisson mixtures with an unknown number of components

In this paper we present Bayesian analysis of finite mixtures of multivariate Poisson distributions with an unknown number of components. The multivariate Poisson distribution can be regarded as the discrete counterpart of the multivariate normal distribution, which is suitable for modelling multivariate count data. Mixtures of multivariate Poisson distributions allow for overdispersion and for negative correlations between variables. To perform Bayesian analysis of these models we adopt a reversible jump Markov chain Monte Carlo (MCMC) algorithm with birth and death moves for updating the number of components. We present results obtained from applying our modelling approach to simulated and real data. Furthermore, we apply our approach to a problem in multivariate disease mapping, namely joint modelling of diseases with correlated counts.     

M-estimator-based robust estimation of the number of components of a superimposed sinusoidal signal model

In this paper, we consider the problem of estimating the number of components of a superimposed nonlinear sinusoids model of a signal in the presence of additive noise. We propose and provide a detailed empirical comparison of robust methods for estimation of the number of components. The proposed methods, which are robust modifications of the commonly used information theoretic criteria, are based on various M-estimator approaches and are robust with respect to outliers present in the data and heavy-tailed noise. The proposed methods are compared with the usual non-robust methods through extensive simulations under varied model scenarios. We also present real signal analysis of two speech signals to show the usefulness of the proposed methodology.     

On the mean past lifetime of the components of a parallel system

In the study of reliability of the technical systems and subsystems, parallel systems play a very important role. In the present paper, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. It is assumed that at time t the system has failed. Under these conditions, we obtain the mean past lifetime (MPL) of the components of the system. Some properties of MPL are studied. It is shown that the underlying distribution function F can be recovered from the proposed MPL. Also, a comparison between two parallel systems are made based on their MPLs in the case where the components of the system are ordered in terms of reversed hazard rate. Finally a characterization of the uniform distribution is given based on MPL.     

The efficient cross-validation of principal components applied to principal component regression

The cross-validation of principal components is a problem that occurs in many applications of statistics. The naive approach of omitting each observation in turn and repeating the principal component calculations is computationally costly. In this paper we present an efficient approach to leave-one-out cross-validation of principal components. This approach exploits the regular nature of leave-one-out principal component eigenvalue downdating. We derive influence statistics and consider the application to principal component regression.     

Non-negative estimation of variance components in heteroscedastic one-way random-effects ANOVA models

There is a considerable amount of literature dealing with inference about the parameters in a heteroscedastic one-way random-effects ANOVA model. In this paper, we primarily address the problem of improved quadratic estimation of the random-effect variance component. It turns out that such estimators with a smaller mean squared error compared with some standard unbiased quadratic estimators exist under quite general conditions. Improved estimators of the error variance components are also established.     

Ordering conditional general coherent systems with exchangeable components

This paper investigates some ordering properties of the residual lives and the inactivity times of coherent systems with dependent exchangeable absolutely continuous components, based on the stochastically ordered signatures between systems, extending the results of Li and Zhang [2008. Some stochastic comparisons of conditional coherent systems. Applied Stochastic Models in Business and Industry 24, 541–549] for the case of independent and identically distributed components.     

Reliability modeling of systems with two dependent degrading components based on gamma processes

Abstract

Many engineering systems have multiple components with more than one degradation measure which is dependent on each other due to their complex failure mechanisms, which results in some insurmountable difficulties for reliability work in engineering. To overcome these difficulties, the system reliability prediction approaches based on performance degradation theory develop rapidly in recent years, and show their superiority over the traditional approaches in many applications. This paper proposes reliability models of systems with two dependent degrading components. It is assumed that the degradation paths of the components are governed by gamma processes. For a parallel system, its failure probability function can be approximated by the bivariate Birnbaum–Saunders distribution. According to the relationship of parallel and series systems, it is easy to find that the failure probability function of a series system can be expressed by the bivariate Birnbaum–Saunders distribution and its marginal distributions. The model in such a situation is very complicated and analytically intractable, and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum likelihood estimates of the parameters to be determined in an efficient manner. After that, the confidence intervals of the failure probability of systems are given. For an illustration of the proposed model, a numerical example about railway track is presented.     

The Posterior Distribution of the Parameters of Component Lifetimes Based on Autopsy Data in a Shock Model

In this paper we consider a binary, monotone system whose component states are dependent through the possible occurrence of independent common shocks, i.e. shocks that destroy several components at once. The individual failure of a component is also thought of as a shock. Such systems can be used to model common cause failures in reliability analysis. The system may be a technological one, or a human being. It is observed until it fails or dies. At this instant, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. These are the so-called autopsy data of the system. For the case of independent components, i.e. no common shocks, Meilijson (1981), Nowik (1990), Antoine et al . (1993) and GTsemyr (1998) discuss the corresponding identifiability problem, i.e. whether the component life distributions can be determined from the distribution of the observed data. Assuming a model where autopsy data is known to be enough for identifia bility, Meilijson (1994) goes beyond the identifiability question and into maximum likelihood estimation of the parameters of the component lifetime distributions based on empirical autopsy data from a sample of several systems. He also considers life-monitoring of some components and conditional life-monitoring of some other. Here a corresponding Bayesian approach is presented for the shock model. Due to prior information one advantage of this approach is that the identifiability problem represents no obstacle. The motivation for introducing the shock model is that the autopsy model is of special importance when components can not be tested separately because it is difficult to reproduce the conditions prevailing in the functioning system. In Gåsemyr & Natvig (1997) we treat the Bayesian approach to life-monitoring and conditional life- monitoring of components     

Ordering and selecting components in multivariate or functional data linear prediction

Summary.  The problem of component choice in regression-based prediction has a long history. The main cases where important choices must be made are functional data analysis, and problems in which the explanatory variables are relatively high dimensional vectors. Indeed, principal component analysis has become the basis for methods for functional linear regression. In this context the number of components can also be interpreted as a smoothing parameter, and so the viewpoint is a little different from that for standard linear regression. However, arguments for and against conventional component choice methods are relevant to both settings and have received significant recent attention. We give a theoretical argument, which is applicable in a wide variety of settings, justifying the conventional approach. Although our result is of minimax type, it is not asymptotic in nature; it holds for each sample size. Motivated by the insight that is gained from this analysis, we give theoretical and numerical justification for cross-validation choice of the number of components that is used for prediction. In particular we show that cross-validation leads to asymptotic minimization of mean summed squared error, in settings which include functional data analysis.     

On the effect of dependency in information properties of series and parallel systems

Information measures of reliability systems has been widely studied in the statistical and reliability literatures. These findings were obtained when lifetimes of components are independent and identically distributed. But, there is no context about the information properties of such systems when lifetimes of components are dependent. In this paper, we explore properties of the entropy and Kullback–Leibler discrimination information for series and parallel system’s lifetimes when lifetimes of components are dependent and identically distributed. Specifically, we give some results on the entropy of series systems when lifetimes of components are positive or negative quadrant dependence. Moreover, several results are obtained about the entropy ordering properties related to other well known stochastic orders. To illustrate the quality of the given results, some examples are also given.     

Study of a Markov model for a high-quality dependent process

For high-quality processes, non-conforming items are seldom observed and the traditional p (or np) charts are not suitable for monitoring the state of the process. A type of chart based on the count of cumulative conforming items has recently been introduced and it is especially useful for automatically collected one-at-a-time data. However, in such a case, it is common that the process characteristics become dependent as items produced one after another are inspected. In this paper, we study the problem of process monitoring when the process is of high quality and measurement values possess a certain serial dependence. The problem of assuming independence is examined and a Markov model for this type of process is studied, upon which suitable control procedures can be developed.     

Modeling systems with two dependent components under bivariate shock models

Series and parallel systems consisting of two dependent components are studied under bivariate shock models. The random variables N₁ and N₂ that represent respectively the number of shocks until failure of component 1 and component 2 are assumed to be dependent and phase-type. The times between successive shocks are assumed to follow a continuous phase-type distribution, and survival functions and mean time to failure values of series and parallel systems are obtained in matrix forms. An upper bound for the joint survival function of the components is also provided under the particular case when the times between shocks follow exponential distribution.     

Generalized likelihood-ratio test of the number of components in finite mixture models

The number of components is an important feature in finite mixture models. Because of the irregularity of the parameter space, the log-likelihood-ratio statistic does not have a chi-square limit distribution. It is very difficult to find a test with a specified significance level, and this is especially true for testing k — 1 versus k components. Most of the existing work has concentrated on finding a comparable approximation to the limit distribution of the log-likelihood-ratio statistic. In this paper, we use a statistic similar to the usual log likelihood ratio, but its null distribution is asymptotically normal. A simulation study indicates that the method has good power at detecting extra components. We also discuss how to improve the power of the test, and some simulations are performed.     

Ordering Properties of Systems with Two Dependent Components

The study of systems with dependent components from a reliability point of view is a very important topic. However, the majority of the articles study the case of independent components. In this article, we study how the dependency influences the performance of the system. We extend some comparison results obtained in the case of independent components to the case of two dependent components. We show that the more diverse the exponential parameters of the two components, the stronger (weaker) the parallel (series) system in the stochastic ordering. We apply our general results to some common bivariate models in the reliability theory.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.